In this section, we briefly review the ASL method and then present the theory that underpins our new method of determining relative source locations from seismic amplitudes.

### ASL method

When body waves propagate, the observed seismic amplitude \(A^{(i)}(f)\) at a certain frequency *f* at the *i*th station can be represented as

$$\begin{aligned} A^{(i)}(f) = A^{(s)}(f) \frac{\exp (-B(f) r^{(i)})}{r^{(i)}} S^{(i)}(f), \end{aligned}$$

(1)

where \(A^{(s)}(f)\) is the source radiation amplitude, \(r^{(i)}\) the hypocentral distance between the source and the *i*th station, and \(S^{(i)}(f)\) the site amplification factor at the *i*th station. *B*(*f*) is defined as

$$\begin{aligned} B(f) = \frac{\pi f}{Q \beta }, \end{aligned}$$

(2)

where *Q* is the intrinsic attenuation factor and \(\beta\) the velocity of the medium, or the average S-wave velocity in general. If we assume the source location for a certain event, we can calculate the source radiation amplitude \(A^{(s)}(f)\) as

$$\begin{aligned} A^{(s)}(f) = \frac{1}{N} \sum _{i = 1}^{N} \frac{A^{(i)}(f)}{S^{(i)}(f)} r^{(i)} \exp (B(f) r^{(i)}), \end{aligned}$$

(3)

where *N* is the number of observations. Using Eqs. (1) and (3), we calculate the normalized residual *R* as

$$\begin{aligned} R = \frac{\sum _{i = 1}^{N} \{A^{(i)}(f)/S^{(i)}(f) - A^{(s)}(f)\exp (-B(f)r^{(i)})/r^{(i)}\}^2}{\sum _{i = 1}^{N} \{A^{(i)}(f)/S^{(i)}(f)\}^2}. \end{aligned}$$

(4)

In application of the ASL method, a grid search is usually conducted to find the location where *R* reaches its minimum value. Because the heterogeneous radiation pattern of the source is not modeled in Eq. (1), and because distortion of the radiation pattern becomes more obvious in higher frequency ranges (e.g., Takemura et al. 2009; Kobayashi et al. 2015), seismic amplitudes at high frequencies (usually higher than 5 Hz) are suitable for the ASL method.

### Estimation of relative source locations from seismic amplitudes

As shown by Eq. (3), the site amplification factor \(S^{(i)}(f)\) has a large influence on source locations estimated by the ASL method. The coda normalization method (e.g., Phillips and Aki 1986; Takemoto et al. 2012) is widely used to estimate site amplification factors. Calibration of site amplification factors with seismic events for which locations are well constrained (Ichihara and Matsumoto 2017; Walsh et al. 2017; Kumagai et al. 2019) is preferable when estimating source locations. Nevertheless, some uncertainties are unavoidable when estimating site amplification factors. In this study, we propose the use of amplitude ratios to cancel out uncertainties in site amplification factors.

From Eq. (1), the amplitude ratio between two events, the *j*th and *k*th events, at the *i*th station is

$$\begin{aligned} \frac{A^{(i)}_{k}(f)}{A^{(i)}_{j}(f)} = \frac{A^{(s)}_{k}(f)}{A^{(s)}_{j}(f)} \frac{\exp (-B(f)r^{(i)}_{k})}{\exp (-B(f)r^{(i)}_{j})} \frac{r^{(i)}_{j}}{r^{(i)}_{k}}, \end{aligned}$$

(5)

where \(A^{(i)}_{j}(f)\) and \(A^{(i)}_{k}(f)\) are the observed amplitudes of the *j*th and *k*th events at the *i*th station, \(A^{(s)}_{j}(f)\) and \(A^{(s)}_{k}(f)\) are the source radiation amplitudes of the *j*th and *k*th events, respectively, and \(r^{(i)}_{j}\) and \(r^{(i)}_{k}\) are the hypocentral distances from the *i*th station to the *j*th and *k*th events, respectively. We define the difference between the hypocentral distances of the *j*th and *k*th events with respect to the *i*th station \(\Delta r^{(i)}_{jk}\) as

$$\begin{aligned} \Delta r^{(i)}_{jk} = r^{(i)}_{k} - r^{(i)}_{j}, \end{aligned}$$

(6)

and, substituting Eq. (6) into (5) gives

$$\begin{aligned} \frac{A^{(i)}_{k}(f)}{A^{(i)}_{j}(f)} = \frac{A^{(s)}_{k}(f)}{A^{(s)}_{j}(f)} \exp (-B(f) \Delta r^{(i)}_{jk}) \left( 1 + \frac{\Delta r^{(i)}_{jk}}{r^{(i)}_{j}} \right) ^{-1}. \end{aligned}$$

(7)

Here we assume that the sources of the *j*th and *k*th events are near each other so that \(\Delta r^{(i)}_{jk}\) becomes much smaller than the hypocentral distances \(r^{(i)}_{j}\) and \(r^{(i)}_{k}\). Taking the natural logarithm of both sides of Eq. (7) and approximating \(\ln (1 + \Delta r^{(i)}_{jk} / r^{(i)}_{j})\) as \(\Delta r^{(i)}_{jk} / r^{(i)}_{j}\), Eq. (7) becomes

$$\begin{aligned} \ln \frac{A^{(i)}_{k}(f)}{A^{(i)}_{j}(f)} \approx \ln \frac{A^{(s)}_{k}(f)}{A^{(s)}_{j}(f)} - B(f) \Delta r^{(i)}_{jk} - \frac{\Delta r^{(i)}_{jk}}{r^{(i)}_{j}}. \end{aligned}$$

(8)

Similar to the formulation of Aoki (1974) for a master event location method using differences of phase arrival times, we can approximate \(\Delta r^{(i)}_{jk}\) to be

$$\begin{aligned} \Delta r^{(i)}_{jk} \approx {\mathbf {n}}^{(i)}_{j} \cdot \Delta {\mathbf {x}}_{jk}, \end{aligned}$$

(9)

where \({\mathbf {n}}^{(i)}_{j}\) is the unit vector representing the takeoff angle and azimuth from the *j*th event to the *i*th station, and \(\Delta {\mathbf {x}}_{jk}\) is the location vector of the *k*th event relative to the *j*th event. This approximation is valid for direct waves. After substituting Eq. (9) into (8), we rewrite Eq. (8) into the following matrix form for standard least-square inversion:

$$\begin{aligned} {\mathbf {d}} = {\mathbf {G}}{\mathbf {m}}, \end{aligned}$$

(10)

where \({\mathbf {d}}\) represents the data vector consisting of \(\ln (A^{(i)}_{k} / A^{(i)}_{j})\), \({\mathbf {m}}\) is the model vector consisting of \(\ln (A^{(s)}_{k} / A^{(s)}_{j})\) and \(\Delta {\mathbf {x}}_{jk}\), and \({\mathbf {G}}\) is the kernel consisting of the coefficients of the model vector in the right-hand side of Eq. (8). We can solve Eq. (10) by a standard least-squares method if we have more than four observations for each event. Our formulation is essentially the same as that of the master event location method using differences of phase arrival times (Aoki 1974), except we use seismic amplitudes rather than arrival times.

The corresponding model covariance matrix \({\mathbf {S}}_{m}\) is

$$\begin{aligned} {\mathbf {S}}_{m} = \left( {\mathbf {G}}^{T} {\mathbf {S}}_{d}^{-1} {\mathbf {G}} \right) ^{-1}, \end{aligned}$$

(11)

where \({\mathbf {S}}_{d}\) is the data covariance matrix. In the following analysis, we first calculate the variance of data residuals using all events except the reference event. Assuming that errors in the data are independent each other, we construct a diagonal data covariance matrix with the variance of data residuals to obtain \({\mathbf {S}}_{m}\) by Eq. (11). Because we adopt a common value for variance for all data, the estimation errors of relative source locations derived by Eq. (11) become the same for all events.

Ichihara and Matsumoto (2017) estimated relative locations of volcanic tremor events using seismic amplitude ratios. They calculated amplitude ratios among several stations to eliminate the source radiation amplitude term (\(A^{(s)}(f)\) in Eq. 1) and conducted a grid search to find optimal tremor source locations. It appears that the approach of Ichihara and Matsumoto (2017) estimates relative source locations, because the site amplification terms and intrinsic attenuation factor they used were adjusted to the reference source location. Nevertheless, their principle formulation follows the original ASL method. In contrast, we attribute the amplitude ratio between two events at each station to the relative difference between their locations. Our approach relies on the assumption that the difference between the hypocentral distance of a reference event and a subevent is much smaller than their hypocentral distances, which is not required in the formulation of Ichihara and Matsumoto (2017). Our formulation thus has two advantages over theirs. First, we avoid uncertainties in estimating site amplification factors. Second, our fundamental formulation (Eq. 8) has a simple linear form, so that we can estimate not only relative source locations, but also the errors on those estimations using a standard least-squares method.

### Data and analysis

Meakandake volcano (eastern Hokkaido, northern Japan; Fig. 1a) has three active craters: Naka-machineshiri, Pon-machineshiri, and Mt. Akanfuji. The eruptive history of Meakandake (Japan Meteorological Agency 2013) shows that its most recent eruption was a phreatic eruption in November 2008 at the 96-1 crater, on the southeastern edge of the Pon-machineshiri crater (Ishimaru et al. 2009). Many earthquakes and tremors were observed before and during the 2008 eruptive period (Ogiso and Yomogida 2012; Japan Meteorological Agency 2013). In this study, we used seismograms of VT earthquakes and volcanic tremors recorded during the 2008 activity at Meakandake by five stations (Fig. 1a) operated by the Sapporo Regional Volcano Observation and Warning Center (RVOWC) of the Japan Meteorological Agency. Vertical-component short-period (natural period 1 s) seismometers were deployed at stations V.PMNS and V.NSYM, and three-component short-period (1 s) seismometers were deployed at the other three stations. Each seismogram was digitized at a sampling rate of 0.01 s.

In this study, we used 1-D velocity and attenuation structure of S-waves shown in Fig. 1b. The velocity structure was derived by trial-and-error approach at Sapporo RVOWC, which has been used there for routine hypocenter determinations since August 2017 (Okuyama, 2020, personal communication). The attenuation structure we used is that of Kumagai et al. (2019), which they used in their application of the ASL method at Nevado del Ruiz volcano (Colombia). We modified the depth of the boundary of the attenuation structure of Kumagai et al. (2019) to be consistent with the velocity structure at Meakandake volcano. We conducted a ray shooting in a spherical coordinate (e.g., Aki and Richards 1980, Chapter 13) to derive \({\mathbf {n}}^{(i)}_{j}\) in Eq. (9).

To validate the source locations of the VT earthquakes determined by our new method by comparing them with those determined from phase arrival times, we selected 45 earthquake events that occurred near the 96-1 crater between 1 and 10 November 2008 for which the Sapporo RVOWC had determined hypocenters from phase arrival times. We manually picked up the arrival times of P-waves at all five stations and those of S-waves at stations V.MEAB and V.MNDK (Fig. 2). Note that because the phase arrivals for these earthquakes were clearly evident on all of the seismograms, any event location method using phase arrival times would be suitable for further analysis of this seismic activity. We calculated source locations by three methods: (a) absolute hypocenter estimation from phase arrival times with the HYPOMH algorithm (Hirata and Matsu’ura 1987), (b) master event location estimation (Aoki 1974) using differences of P-wave arrival times, (c) ASL method using seismic amplitudes and (d) our new method of estimating relative source locations. The reference event we used for the two methods of estimating relative source locations was an earthquake at 21:30 (Japan Standard Time; JST) on 7 November 2008 (latitude 43.3829°, longitude 144.0093°, depth \(-0.40\) km). This reference location was originally determined from the phase arrival times and the HYPOMH algorithm. We selected this reference event because its epicenter was at the centroid of all of the hypocenters derived from the phase arrival times. The P-wave velocity structure we used for the HYPOMH algorithm and master event method was \(\sqrt{3}\) times larger than S-wave velocity structure shown in Fig. 1b. For the ASL and our relative source location methods, we applied a 5–10 Hz bandpass filter to the data of each seismogram, and calculated the root-mean-square (RMS) amplitude of the vertical component within a time window extending for 10 s after the P-wave arrival (Fig. 2). The time window includes the whole waveform record of the VT earthquake so that direct, narrow- and wide-angle scattered waves affect the RMS amplitude. The amplitude of late-arriving coda waves, that is, that of wide-angle scattered waves, is so small that the direct and narrow-angle scattered waves mainly contribute to the obtained RMS amplitude at each station. Although our formulation presented in the previous section is rigorously valid only for direct waves, the formulation still holds for the RMS amplitudes because the takeoff angle and azimuth of narrow-angle scattered waves can be considered to be very close to those of the direct waves (Sato and Emoto 2018). We set frequency *f* in Eq. (2) to 7.5 Hz. For the ASL method, we used the same amplification factors as those estimated by Ogiso and Yomogida (2012) using the coda normalization method, and performed a grid search with 0.001°increments of latitude and longitude and a 0.1 km depth increment. Because the reference event located on the shallower layer with \(Q = 40\), we used the value in our new method regardless of the two-layer attenuation structure (Fig. 1b).

For comparison of the calculated volcanic tremor locations, we selected tremors that occurred on 16 and 17 November 2008. Ogiso and Yomogida (2012) applied the ASL method to these events and identified segmentation in their source regions. The duration time of 16 November tremor was about 30 min (Ogiso and Yomogida 2012). Following the approach of Ogiso and Yomogida (2012), we divided the first 17 min of the tremor into three phases (Fig. 3a) and estimated source locations for each phase. During 17 and 18 November, small-amplitude, long-duration tremors were observed intermittently (Ogiso and Yomogida 2012, Fig. 18). Because amplitude ratios among stations did not change significantly during these intermittent tremors (Ogiso and Yomogida 2012, Fig. 21), we estimated tremor locations from 11:00 to 12:00 (JST) on 17 November (Fig. 3b). Because the velocity and attenuation structures we used (Fig. 1b) differed from the simple structures used by Ogiso and Yomogida (2012), we analyzed the locations of these tremors by both the ASL method and our new relative source location method. The process we used to prepare amplitude data was similar to that used for analysis of volcanic earthquakes, apart from the length of the time window. After applying a 5–10 Hz bandpass filter, we calculated the time series of RMS amplitudes from the vertical-component seismogram at each station within a 30-s time window that we shifted by 15 s for each calculation. We then used these time series of RMS amplitudes for both the ASL and our relative source location methods. For our relative source location method, we set the reference location for tremors of both 16 and 17 November at latitude 43.378°, longitude 144.005°and 0.1 km depth, which was the tremor location from 01:05:05 to 01:05:35 (JST) on 16 November, as estimated by the ASL method in this study. Same as the case of VT earthquakes, we used \(f=7.5\) (Hz) and \(Q = 40\) in our method. The calculation procedure of the ASL method for the tremors was the same as that for the VT earthquakes.